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Sign-vector determination of k-face Schubert arrangement cells

Prove that, for any convex polytope, the cells in its k-face Schubert arrangement are determined by their sign vectors (Conjecture 3.5).

References

Show that the cells in the $k$-face Schubert arrangement of any convex polytope are determined by their sign vectors. \hfill Conjecture 3.5.

What is Positive Geometry? (Ranestad et al., 18 Feb 2025) in Section “Open questions”: Problem [Seemann]